The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(n^1)).

0 CpxTRS
↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID), 0 ms)
↳2 CdtProblem
    ↳3 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳4 CdtProblem
    ↳5 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳6 CdtProblem
    ↳7 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 5 ms)
    ↳8 CdtProblem
↳9 CpxTrsMatchBoundsTAProof (⇔, 0 ms)
↳10 BOUNDS(O(1), O(n^1))

(0) Obligation:

Runtime Complexity TRS:
The TRS R consists of the following rules:

f(f(a, a), x) → f(a, f(b, f(a, x)))
f(x, f(y, z)) → f(f(x, y), z)

Rewrite Strategy: INNERMOST

(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(f(a, a), z0) → f(a, f(b, f(a, z0)))
f(z0, f(z1, z2)) → f(f(z0, z1), z2)
Tuples:

F(f(a, a), z0) → c(F(a, f(b, f(a, z0))), F(b, f(a, z0)), F(a, z0))
F(z0, f(z1, z2)) → c1(F(f(z0, z1), z2), F(z0, z1))
S tuples:

F(f(a, a), z0) → c(F(a, f(b, f(a, z0))), F(b, f(a, z0)), F(a, z0))
F(z0, f(z1, z2)) → c1(F(f(z0, z1), z2), F(z0, z1))
K tuples:none
Defined Rule Symbols:

f

Defined Pair Symbols:

F

Compound Symbols:

c, c1

(3) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace F(f(a, a), z0) → c(F(a, f(b, f(a, z0))), F(b, f(a, z0)), F(a, z0)) by

F(f(a, a), z2) → c(F(a, f(f(b, a), z2)), F(b, f(a, z2)), F(a, z2))
F(f(a, a), f(z1, z2)) → c(F(a, f(b, f(f(a, z1), z2))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2)))
F(f(a, a), x0) → c

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(f(a, a), z0) → f(a, f(b, f(a, z0)))
f(z0, f(z1, z2)) → f(f(z0, z1), z2)
Tuples:

F(z0, f(z1, z2)) → c1(F(f(z0, z1), z2), F(z0, z1))
F(f(a, a), z2) → c(F(a, f(f(b, a), z2)), F(b, f(a, z2)), F(a, z2))
F(f(a, a), f(z1, z2)) → c(F(a, f(b, f(f(a, z1), z2))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2)))
F(f(a, a), x0) → c
S tuples:

F(z0, f(z1, z2)) → c1(F(f(z0, z1), z2), F(z0, z1))
F(f(a, a), z2) → c(F(a, f(f(b, a), z2)), F(b, f(a, z2)), F(a, z2))
F(f(a, a), f(z1, z2)) → c(F(a, f(b, f(f(a, z1), z2))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2)))
F(f(a, a), x0) → c
K tuples:none
Defined Rule Symbols:

f

Defined Pair Symbols:

F

Compound Symbols:

c1, c, c

(5) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

Removed 1 trailing nodes:

F(f(a, a), x0) → c

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(f(a, a), z0) → f(a, f(b, f(a, z0)))
f(z0, f(z1, z2)) → f(f(z0, z1), z2)
Tuples:

F(z0, f(z1, z2)) → c1(F(f(z0, z1), z2), F(z0, z1))
F(f(a, a), z2) → c(F(a, f(f(b, a), z2)), F(b, f(a, z2)), F(a, z2))
F(f(a, a), f(z1, z2)) → c(F(a, f(b, f(f(a, z1), z2))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2)))
S tuples:

F(z0, f(z1, z2)) → c1(F(f(z0, z1), z2), F(z0, z1))
F(f(a, a), z2) → c(F(a, f(f(b, a), z2)), F(b, f(a, z2)), F(a, z2))
F(f(a, a), f(z1, z2)) → c(F(a, f(b, f(f(a, z1), z2))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2)))
K tuples:none
Defined Rule Symbols:

f

Defined Pair Symbols:

F

Compound Symbols:

c1, c

(7) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace F(f(a, a), f(z1, z2)) → c(F(a, f(b, f(f(a, z1), z2))), F(b, f(a, f(z1, z2))), F(a, f(z1, z2))) by

F(f(a, a), f(x0, z2)) → c(F(a, f(f(b, f(a, x0)), z2)), F(b, f(a, f(x0, z2))), F(a, f(x0, z2)))
F(f(a, a), f(a, z0)) → c(F(a, f(b, f(a, f(b, f(a, z0))))), F(b, f(a, f(a, z0))), F(a, f(a, z0)))
F(f(a, a), f(f(z1, z2), x1)) → c(F(a, f(b, f(f(f(a, z1), z2), x1))), F(b, f(a, f(f(z1, z2), x1))), F(a, f(f(z1, z2), x1)))
F(f(a, a), f(x0, x1)) → c(F(b, f(a, f(x0, x1))), F(a, f(x0, x1)))

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(f(a, a), z0) → f(a, f(b, f(a, z0)))
f(z0, f(z1, z2)) → f(f(z0, z1), z2)
Tuples:

F(z0, f(z1, z2)) → c1(F(f(z0, z1), z2), F(z0, z1))
F(f(a, a), z2) → c(F(a, f(f(b, a), z2)), F(b, f(a, z2)), F(a, z2))
F(f(a, a), f(x0, z2)) → c(F(a, f(f(b, f(a, x0)), z2)), F(b, f(a, f(x0, z2))), F(a, f(x0, z2)))
F(f(a, a), f(a, z0)) → c(F(a, f(b, f(a, f(b, f(a, z0))))), F(b, f(a, f(a, z0))), F(a, f(a, z0)))
F(f(a, a), f(f(z1, z2), x1)) → c(F(a, f(b, f(f(f(a, z1), z2), x1))), F(b, f(a, f(f(z1, z2), x1))), F(a, f(f(z1, z2), x1)))
F(f(a, a), f(x0, x1)) → c(F(b, f(a, f(x0, x1))), F(a, f(x0, x1)))
S tuples:

F(z0, f(z1, z2)) → c1(F(f(z0, z1), z2), F(z0, z1))
F(f(a, a), z2) → c(F(a, f(f(b, a), z2)), F(b, f(a, z2)), F(a, z2))
F(f(a, a), f(x0, z2)) → c(F(a, f(f(b, f(a, x0)), z2)), F(b, f(a, f(x0, z2))), F(a, f(x0, z2)))
F(f(a, a), f(a, z0)) → c(F(a, f(b, f(a, f(b, f(a, z0))))), F(b, f(a, f(a, z0))), F(a, f(a, z0)))
F(f(a, a), f(f(z1, z2), x1)) → c(F(a, f(b, f(f(f(a, z1), z2), x1))), F(b, f(a, f(f(z1, z2), x1))), F(a, f(f(z1, z2), x1)))
F(f(a, a), f(x0, x1)) → c(F(b, f(a, f(x0, x1))), F(a, f(x0, x1)))
K tuples:none
Defined Rule Symbols:

f

Defined Pair Symbols:

F

Compound Symbols:

c1, c, c

(9) CpxTrsMatchBoundsTAProof (EQUIVALENT transformation)

A linear upper bound on the runtime complexity of the TRS R could be shown with a Match-Bound[TAB_LEFTLINEAR,TAB_NONLEFTLINEAR] (for contructor-based start-terms) of 0.

The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by:
final states : [1]
transitions:
a0() → 0
b0() → 0
f0(0, 0) → 1

(10) BOUNDS(O(1), O(n^1))